Gauss’ Law

Contents

1 Gauss’s law 11.1 Qualitative description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Equation involving E field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 Integral form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Differential form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Equivalence of integral and differential forms . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Equation involving D field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.1 Free, bound, and total charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Integral form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.3 Differential form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Equivalence of total and free charge statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Equation for linear materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 Relation to Coulomb’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.6.1 Deriving Gauss’s law from Coulomb’s law . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6.2 Deriving Coulomb’s law from Gauss’s law . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Electric flux 42.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Ampère’s circuital law 53.1 Ampère’s original circuital law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1.1 Integral form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.1.2 Differential form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Note on free current versus bound current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Shortcomings of the original formulation of Ampère’s circuital law . . . . . . . . . . . . . . . . . . 6

3.3.1 Displacement current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 Extending the original law: the Maxwell–Ampère equation . . . . . . . . . . . . . . . . . . . . . . 7

i

ii CONTENTS

3.4.1 Proof of equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.5 Ampère’s law in cgs units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Divergence theorem 104.1 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Mathematical statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2.1 Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.4.1 Differential form and integral form of physical laws . . . . . . . . . . . . . . . . . . . . . 124.4.2 Inverse-square laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.7.1 Multiple dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.7.2 Tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Electric displacement field 155.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3 Example: Displacement field in a capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.6 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Chapter 1

Gauss’s law

This article is about Gauss’s law concerning the elec-tric field. For analogous law concerning differentfields, see Gauss’s law for magnetism and Gauss’s lawfor gravity. For Gauss’s theorem, a mathematical theo-rem relevant to all of these laws, see Divergence theorem.

In physics,Gauss’s law, also known asGauss’s flux the-orem, is a law relating the distribution of electric chargeto the resulting electric field.The law was formulated by Carl Friedrich Gauss in 1835,but was not published until 1867.[1] It is one of Maxwell’sfour equations, which form the basis of classical elec-trodynamics, the other three being Gauss’s law for mag-netism, Faraday’s law of induction, and Ampère’s lawwith Maxwell’s correction. Gauss’s law can be used toderive Coulomb’s law,[2] and vice versa.

1.1 Qualitative description

In words, Gauss’s law states that:

The net electric flux through any closed surfaceis equal to 1⁄ε times the net electric charge en-closed within that closed surface.[3]

Gauss’s law has a close mathematical similarity with anumber of laws in other areas of physics, such as Gauss’slaw for magnetism and Gauss’s law for gravity. In fact,any "inverse-square law" can be formulated in a way sim-ilar to Gauss’s law: For example, Gauss’s law itself is es-sentially equivalent to the inverse-square Coulomb’s law,and Gauss’s law for gravity is essentially equivalent to theinverse-square Newton’s law of gravity.Gauss’s law is something of an electrical analogue ofAmpère’s law, which deals with magnetism.The law can be expressed mathematically using vectorcalculus in integral form and differential form, both areequivalent since they are related by the divergence theo-rem, also called Gauss’s theorem. Each of these forms inturn can also be expressed two ways: In terms of a re-lation between the electric field E and the total electric

charge, or in terms of the electric displacement field Dand the free electric charge.[4]

1.2 Equation involving E field

Gauss’s law can be stated using either the electric field Eor the electric displacement field D. This section showssome of the forms with E; the form with D is below, asare other forms with E.

1.2.1 Integral form

Gauss’s law may be expressed as:[5]

ΦE =Q

ε0

where ΦE is the electric flux through a closed surface Senclosing any volume V, Q is the total charge enclosedwithin S, and ε0 is the electric constant. The electric fluxΦE is defined as a surface integral of the electric field:

ΦE = S E · dA

where E is the electric field, dA is a vector representingan infinitesimal element of area,[note 1] and · represents thedot product of two vectors.Since the flux is defined as an integral of the electric field,this expression of Gauss’s law is called the integral form.

Applying the integral form

Main article: Gaussian surfaceSee also Capacitance (Gauss’s law)

If the electric field is known everywhere, Gauss’s lawmakes it quite easy, in principle, to find the distributionof electric charge: The charge in any given region can bededuced by integrating the electric field to find the flux.

1

2 CHAPTER 1. GAUSS’S LAW

However, much more often, it is the reverse problem thatneeds to be solved: The electric charge distribution isknown, and the electric field needs to be computed. Thisis much more difficult, since if you know the total fluxthrough a given surface, that gives almost no informationabout the electric field, which (for all you know) could goin and out of the surface in arbitrarily complicated pat-terns.An exception is if there is some symmetry in the situation,which mandates that the electric field passes through thesurface in a uniform way. Then, if the total flux is known,the field itself can be deduced at every point. Commonexamples of symmetries which lend themselves to Gauss’slaw include cylindrical symmetry, planar symmetry, andspherical symmetry. See the article Gaussian surface forexamples where these symmetries are exploited to com-pute electric fields.

1.2.2 Differential form

By the divergence theorem, Gauss’s law can alternativelybe written in the differential form:

∇ · E =ρ

ε0

where ∇ ·E is the divergence of the electric field, ε0 is theelectric constant, and ρ is the total electric charge density(charge per unit volume).

1.2.3 Equivalence of integral and differen-tial forms

Main article: Divergence theorem

The integral and differential forms are mathematicallyequivalent, by the divergence theorem. Here is the ar-gument more specifically.

1.3 Equation involving D field

See also: Maxwell’s equations

1.3.1 Free, bound, and total charge

Main article: Electric polarization

The electric charge that arises in the simplest textbook sit-uations would be classified as “free charge”—for exam-ple, the charge which is transferred in static electricity,

or the charge on a capacitor plate. In contrast, “boundcharge” arises only in the context of dielectric (polariz-able) materials. (All materials are polarizable to someextent.) When such materials are placed in an externalelectric field, the electrons remain bound to their respec-tive atoms, but shift a microscopic distance in responseto the field, so that they're more on one side of the atomthan the other. All these microscopic displacements addup to give a macroscopic net charge distribution, and thisconstitutes the “bound charge”.Althoughmicroscopically, all charge is fundamentally thesame, there are often practical reasons for wanting to treatbound charge differently from free charge. The result isthat the more “fundamental” Gauss’s law, in terms of E(above), is sometimes put into the equivalent form below,which is in terms of D and the free charge only.

1.3.2 Integral form

This formulation of Gauss’s law states the total chargeform:

ΦD = Qfree

where ΦD is the D-field flux through a surface S whichencloses a volumeV, andQ ᵣₑₑ is the free charge containedin V. The flux ΦD is defined analogously to the flux ΦEof the electric field E through S:

ΦD = S D · dA

1.3.3 Differential form

The differential form of Gauss’s law, involving freecharge only, states:

∇ · D = ρfree

where ∇ ·D is the divergence of the electric displacementfield, and ρ ᵣₑₑ is the free electric charge density.

1.4 Equivalence of total and freecharge statements

1.5 Equation for linear materials

In homogeneous, isotropic, nondispersive, linear materi-als, there is a simple relationship between E and D:

1.8. NOTES 3

D = εE

where ε is the permittivity of the material. For the caseof vacuum (aka free space), ε = ε0. Under these circum-stances, Gauss’s law modifies to

ΦE =Qfreeε

for the integral form, and

∇ · E =ρfreeε

for the differential form.

1.6 Relation to Coulomb’s law

1.6.1 Deriving Gauss’s law fromCoulomb’s law

Gauss’s law can be derived from Coulomb’s law.

Note that since Coulomb’s law only applies to stationarycharges, there is no reason to expect Gauss’s law to holdfor moving charges based on this derivation alone. In fact,Gauss’s law does hold for moving charges, and in this re-spect Gauss’s law is more general than Coulomb’s law.

1.6.2 Deriving Coulomb’s law fromGauss’s law

Strictly speaking, Coulomb’s law cannot be derived fromGauss’s law alone, since Gauss’s law does not give anyinformation regarding the curl of E (see Helmholtz de-composition and Faraday’s law). However, Coulomb’slaw can be proven from Gauss’s law if it is assumed,in addition, that the electric field from a point charge isspherically-symmetric (this assumption, like Coulomb’slaw itself, is exactly true if the charge is stationary, andapproximately true if the charge is in motion).

1.7 See also

• Method of image charges

• Uniqueness theorem for Poisson’s equation

1.8 Notes[1] More specifically, the infinitesimal area is thought of as

planar and with area dA. The vector dA is normal to thisarea element and has magnitude dA.[6]

1.9 References[1] Bellone, Enrico (1980). A World on Paper: Studies on the

Second Scientific Revolution.

[2] Halliday, David; Resnick, Robert (1970). Fundamentalsof Physics. John Wiley & Sons, Inc. pp. 452–53.

[3] Serway, Raymond A. (1996). Physics for Scientists andEngineers with Modern Physics, 4th edition. p. 687.

[4] I.S. Grant, W.R. Phillips (2008). Electromagnetism (2nded.). Manchester Physics, JohnWiley & Sons. ISBN 978-0-471-92712-9.

[5] I.S. Grant, W.R. Phillips (2008). Electromagnetism (2nded.). Manchester Physics, JohnWiley & Sons. ISBN 978-0-471-92712-9.

[6] Matthews, Paul (1998). Vector Calculus. Springer. ISBN3-540-76180-2.

[7] See, for example, Griffiths, David J. (2013). Introductionto Electrodynamics (4th ed.). Prentice Hall. p. 50.

Jackson, John David (1998). Classical Electrodynamics,3rd ed., New York: Wiley. ISBN 0-471-30932-X.

1.10 External links• MIT Video Lecture Series (30 x 50 minutelectures)- Electricity and Magnetism Taught by Pro-fessor Walter Lewin.

• section on Gauss’s law in an online textbook

• MISN-0-132 Gauss’s Law for Spherical Symmetry(PDF file) by Peter Signell for Project PHYSNET.

• MISN-0-133 Gauss’s Law Applied to Cylindrical andPlanar Charge Distributions (PDF file) by PeterSignell for Project PHYSNET.

Chapter 2

Electric flux

In electromagnetism, electric flux is the measure of flowof the electric field through a given area. Electric fluxis proportional to the number of electric field lines goingthrough a normally perpendicular surface. If the electricfield is uniform, the electric flux passing through a surfaceof vector area S is

ΦE = E · S = ES cos θ,

where E is the electric field (having units of V/m), E is itsmagnitude, S is the area of the surface, and θ is the anglebetween the electric field lines and the normal (perpen-dicular) to S.For a non-uniform electric field, the electric flux dΦEthrough a small surface area dS is given by

dΦE = E · dS

(the electric field, E, multiplied by the component of areaperpendicular to the field). The electric flux over a surfaceS is therefore given by the surface integral:

ΦE =

∫∫S

E · dS

where E is the electric field and dS is a differential areaon the closed surface S with an outward facing surfacenormal defining its direction.For a closed Gaussian surface, electric flux is given by:

ΦE = S E · dS = Qϵ0

where

E is the electric field,S is any closed surface,Q is the total electric charge inside the surfaceS,

ε0 is the electric constant (a universal constant,also called the "permittivity of free space”) (ε0≈ 8.854 187 817... x 10−12 farads per meter(F·m−1)).

This relation is known as Gauss’ law for electric field inits integral form and it is one of the four Maxwell’s equa-tions.While the electric flux is not affected by charges that arenot within the closed surface, the net electric field, E, inthe Gauss’ Law equation, can be affected by charges thatlie outside the closed surface. While Gauss’ Law holds forall situations, it is only useful for “by hand” calculationswhen high degrees of symmetry exist in the electric field.Examples include spherical and cylindrical symmetry.Electrical flux has SI units of volt metres (Vm), or, equiv-alently, newton metres squared per coulomb (N m2 C−1).Thus, the SI base units of electric flux are kg·m3·s−3·A−1.Its dimensional formula is [L3MT−3I−1].

2.1 See also• Magnetic flux

• Maxwell’s equations

related websites are following: http://www.citycollegiate.com/coulomb4_XII.htm[1]

2.2 References[1] electric flux

2.3 External links• Electric flux — HyperPhysics

4

Chapter 3

Ampère’s circuital law

“Ampère’s law” redirects here. For the law describingforces between current-carrying wires, see Ampère’sforce law.

In classical electromagnetism, Ampère’s circuital law,discovered by André-Marie Ampère in 1826,[1] relatesthe integrated magnetic field around a closed loop to theelectric current passing through the loop. James ClerkMaxwell derived it again using hydrodynamics in his 1861paper On Physical Lines of Force and it is now one ofthe Maxwell equations, which form the basis of classicalelectromagnetism.

3.1 Ampère’s original circuital law

Ampère’s law relates magnetic fields to electric currentsthat produce them. Ampère’s law determines the mag-netic field associated with a given current, or the currentassociated with a given magnetic field, provided that theelectric field does not change over time. In its originalform, Ampère’s circuital law relates a magnetic field toits electric current source. The law can be written in twoforms, the “integral form” and the “differential form”.The forms are equivalent, and related by the Kelvin–Stokes theorem. It can also be written in terms of eitherthe B or H magnetic fields. Again, the two forms areequivalent (see the "proof" section below).Ampère’s circuital law is now known to be a correctlaw of physics in a magnetostatic situation: The systemis static except possibly for continuous steady currentswithin closed loops. In all other cases the law is incor-rect unless Maxwell’s correction is included (see below).

3.1.1 Integral form

In SI units (cgs units are later), the “integral form”of the original Ampère’s circuital law is a line inte-gral of the magnetic field around some closed curve C(arbitrary but must be closed). The curve C in turnbounds both a surface S which the electric current passesthrough (again arbitrary but not closed—since no three-dimensional volume is enclosed by S), and encloses the

current. The mathematical statement of the law is a re-lation between the total amount of magnetic field aroundsome path (line integral) due to the current which passesthrough that enclosed path (surface integral). It can bewritten in a number of forms.[2][3]

In terms of total current, which includes both free andbound current, the line integral of the magnetic B-field(in tesla, T) around closed curve C is proportional to thetotal current Iₑ passing through a surface S (enclosed byC):

∮C

B · dℓ = µ0

∫∫S

J · dS = µ0Ienc

where J is the total current density (in ampere per squaremetre, Am−2).Alternatively in terms of free current, the line integral ofthe magnetic H-field (in ampere per metre, Am−1) aroundclosed curve C equals the free current I , ₑ through asurface S:

∮C

H · dℓ =

∫∫S

Jf · dS = If,enc

where J is the free current density only. Furthermore

• ∮Cis the closed line integral around the closed curve

C,

• ∫∫Sdenotes a 2d surface integral over S enclosed by

C

• • is the vector dot product,

• dℓ is an infinitesimal element (a differential) of thecurve C (i.e. a vector with magnitude equal to thelength of the infinitesimal line element, and direc-tion given by the tangent to the curve C)

• dS is the vector area of an infinitesimal element ofsurface S (that is, a vector with magnitude equal tothe area of the infinitesimal surface element, and di-rection normal to surface S. The direction of the nor-mal must correspond with the orientation ofC by theright hand rule), see below for further explanation ofthe curve C and surface S.

5

6 CHAPTER 3. AMPÈRE’S CIRCUITAL LAW

TheB andH fields are related by the constitutive equation

B = µ0H

where μ0 is the magnetic constant.There are a number of ambiguities in the above definitionsthat require clarification and a choice of convention.

1. First, three of these terms are associated with signambiguities: the line integral

∮Ccould go around the

loop in either direction (clockwise or counterclock-wise); the vector area dS could point in either of thetwo directions normal to the surface; and Iₑ is thenet current passing through the surface S, meaningthe current passing through in one direction, minusthe current in the other direction—but either direc-tion could be chosen as positive. These ambiguitiesare resolved by the right-hand rule: With the palmof the right-hand toward the area of integration, andthe index-finger pointing along the direction of line-integration, the outstretched thumb points in the di-rection that must be chosen for the vector area dS.Also the current passing in the same direction as dSmust be counted as positive. The right hand grip rulecan also be used to determine the signs.

2. Second, there are infinitely many possible surfacesS that have the curve C as their border. (Imagine asoap film on a wire loop, which can be deformed bymoving the wire). Which of those surfaces is to bechosen? If the loop does not lie in a single plane, forexample, there is no one obvious choice. The answeris that it does not matter; it can be proven that anysurface with boundary C can be chosen.

3.1.2 Differential form

By the Stokes’ theorem, this equation can also be writtenin a “differential form”. Again, this equation only appliesin the case where the electric field is constant in time,meaning the currents are steady (time-independent, elsethe magnetic field would change with time); see below forthe more general form. In SI units, the equation states fortotal current:

∇× B = µ0J

and for free current

∇×H = Jf

where ∇× is the curl operator.

3.2 Note on free current versusbound current

The electric current that arises in the simplest textbooksituations would be classified as “free current”—for ex-ample, the current that passes through a wire or battery.In contrast, “bound current” arises in the context of bulkmaterials that can be magnetized and/or polarized. (Allmaterials can to some extent.)When a material is magnetized (for example, by placing itin an external magnetic field), the electrons remain boundto their respective atoms, but behave as if they were orbit-ing the nucleus in a particular direction, creating a micro-scopic current. When the currents from all these atomsare put together, they create the same effect as a macro-scopic current, circulating perpetually around the magne-tized object. This magnetization current JM is one con-tribution to “bound current”.The other source of bound current is bound charge. Whenan electric field is applied, the positive and negative boundcharges can separate over atomic distances in polarizablematerials, and when the bound charges move, the po-larization changes, creating another contribution to the“bound current”, the polarization current JP.The total current density J due to free and bound chargesis then:

J = Jf + JM + JPwith J the “free” or “conduction” current density.All current is fundamentally the same, microscopically.Nevertheless, there are often practical reasons for want-ing to treat bound current differently from free current.For example, the bound current usually originates overatomic dimensions, and one may wish to take advantageof a simpler theory intended for larger dimensions. Theresult is that the more microscopic Ampère’s law, ex-pressed in terms of B and the microscopic current (whichincludes free, magnetization and polarization currents), issometimes put into the equivalent form below in termsof H and the free current only. For a detailed definitionof free current and bound current, and the proof that thetwo formulations are equivalent, see the "proof" sectionbelow.

3.3 Shortcomings of the originalformulation of Ampère’s cir-cuital law

There are two important issues regarding Ampère’s lawthat require closer scrutiny. First, there is an issue re-garding the continuity equation for electrical charge. Invector calculus, the identity for the divergence of a curl

3.4. EXTENDING THE ORIGINAL LAW: THE MAXWELL–AMPÈRE EQUATION 7

states that a vector field’s curl divergence must always bezero. Hence

∇ · (∇× B) = 0

and so the original Ampère’s law implies that

∇ · J = 0.

But in general

∇ · J = −∂ρ

∂t

which is non-zero for a time-varying charge density. Anexample occurs in a capacitor circuit where time-varyingcharge densities exist on the plates.[4][5][6][7][8]

Second, there is an issue regarding the propagation ofelectromagnetic waves. For example, in free space, where

J = 0,

Ampère’s law implies that

∇× B = 0

but instead

∇× B =1

c2∂E∂t

.

To treat these situations, the contribution of displacementcurrent must be added to the current term in Ampère’slaw.James Clerk Maxwell conceived of displacement currentas a polarization current in the dielectric vortex sea, whichhe used to model the magnetic field hydrodynamicallyand mechanically.[9] He added this displacement currentto Ampère’s circuital law at equation (112) in his 1861paper On Physical Lines of Force.[10]

3.3.1 Displacement current

Main article: Displacement current

In free space, the displacement current is related to thetime rate of change of electric field.In a dielectric the above contribution to displacement cur-rent is present too, but a major contribution to the dis-placement current is related to the polarization of theindividual molecules of the dielectric material. Even

though charges cannot flow freely in a dielectric, thecharges in molecules can move a little under the influenceof an electric field. The positive and negative chargesin molecules separate under the applied field, causingan increase in the state of polarization, expressed as thepolarization density P. A changing state of polarization isequivalent to a current.Both contributions to the displacement current are com-bined by defining the displacement current as:[4]

JD =∂

∂tD(r, t) ,

where the electric displacement field is defined as:

D = ε0E+ P = ε0εrE ,

where ε0 is the electric constant, εᵣ the relative static per-mittivity, and P is the polarization density. Substitutingthis form for D in the expression for displacement cur-rent, it has two components:

JD = ε0∂E∂t

+∂P∂t

.

The first term on the right hand side is present every-where, even in a vacuum. It doesn't involve any actualmovement of charge, but it nevertheless has an associ-ated magnetic field, as if it were an actual current. Someauthors apply the name displacement current to only thiscontribution.[11]

The second term on the right hand side is the displace-ment current as originally conceived by Maxwell, associ-ated with the polarization of the individual molecules ofthe dielectric material.Maxwell’s original explanation for displacement currentfocused upon the situation that occurs in dielectric me-dia. In the modern post-aether era, the concept hasbeen extended to apply to situations with no materialmedia present, for example, to the vacuum between theplates of a charging vacuum capacitor. The displace-ment current is justified today because it serves severalrequirements of an electromagnetic theory: correct pre-diction of magnetic fields in regions where no free currentflows; prediction of wave propagation of electromagneticfields; and conservation of electric charge in cases wherecharge density is time-varying. For greater discussion seeDisplacement current.

3.4 Extending the original law: theMaxwell–Ampère equation

Next Ampère’s equation is extended by including the po-larization current, thereby remedying the limited applica-bility of the original Ampère’s circuital law.

8 CHAPTER 3. AMPÈRE’S CIRCUITAL LAW

Treating free charges separately from bound charges,Ampère’s equation including Maxwell’s correction interms of the H-field is (the H-field is used because it in-cludes the magnetization currents, so JM does not appearexplicitly, see H-field and also Note):[12]

∮C

H · dℓ =

∫∫S

(Jf +

∂

∂tD)· dS

(integral form), where H is the magnetic H field (alsocalled “auxiliary magnetic field”, “magnetic field inten-sity”, or just “magnetic field”), D is the electric displace-ment field, and J is the enclosed conduction current orfree current density. In differential form,

∇×H = Jf +∂

∂tD .

On the other hand, treating all charges on the samefooting (disregarding whether they are bound or freecharges), the generalized Ampère’s equation, also calledthe Maxwell–Ampère equation, is in integral form (seethe "proof" section below):

In differential form,

In both forms J includes magnetization current density[13]as well as conduction and polarization current densi-ties. That is, the current density on the right side of theAmpère–Maxwell equation is:

Jf + JD + JM = Jf + JP + JM + ε0∂E∂t

= J+ ε0∂E∂t

,

where current density JD is the displacement current, andJ is the current density contribution actually due to move-ment of charges, both free and bound. Because∇ ·D = ρ,the charge continuity issue with Ampère’s original formu-lation is no longer a problem.[14] Because of the term inε0∂E / ∂t, wave propagation in free space now is possible.With the addition of the displacement current, Maxwellwas able to hypothesize (correctly) that light was a formof electromagnetic wave. See electromagnetic waveequation for a discussion of this important discovery.

3.4.1 Proof of equivalence

3.5 Ampère’s law in cgs units

In cgs units, the integral form of the equation, includingMaxwell’s correction, reads

∮C

B · dℓ =1

c

∫∫S

(4πJ+ ∂E

∂t

)· dS

where c is the speed of light.The differential form of the equation (again, includingMaxwell’s correction) is

∇× B =1

c

(4πJ+ ∂E

∂t

).

3.6 See also

3.7 Notes[1] Richard Fitzpatrick (2007). “Ampère’s Circuital Law”.

[2] Heinz E Knoepfel (2000). Magnetic Fields: A compre-hensive theoretical treatise for practical use. Wiley. p. 4.ISBN 0-471-32205-9.

[3] George E. Owen (2003). Electromagnetic Theory (Reprintof 1963 ed.). Courier-Dover Publications. p. 213. ISBN0-486-42830-3.

[4] John David Jackson (1999). Classical Electrodynamics(3rd ed.). Wiley. p. 238. ISBN 0-471-30932-X.

[5] David J Griffiths (1999). Introduction to Electrodynamics(3rd ed.). Pearson/Addison-Wesley. pp. 322–323. ISBN0-13-805326-X.

[6] George E. Owen (2003). op. cit.. Mineola, N.Y.: DoverPublications. p. 285. ISBN 0-486-42830-3.

[7] J. Billingham, A. C. King (2006). Wave Motion. Cam-bridge University Press. p. 179. ISBN 0-521-63450-4.

[8] J.C. Slater and N.H. Frank (1969). Electromagnetism(Reprint of 1947 ed.). Courier Dover Publications. p.83. ISBN 0-486-62263-0.

[9] Daniel M. Siegel (2003). Innovation in Maxwell’s Electro-magnetic Theory: Molecular Vortices, Displacement Cur-rent, and Light. Cambridge University Press. pp. 96–98.ISBN 0-521-53329-5.

[10] James C. Maxwell (1861). “On Physical Lines of Force”(PDF). Philosophical Magazine and Journal of Science.

[11] For example, see David J. Griffiths (1999). op. cit.. Up-per Saddle River, NJ: Prentice Hall. p. 323. ISBN 0-13-805326-X. and Tai L. Chow (2006). Introduction toElectromagnetic Theory. Jones & Bartlett. p. 204. ISBN0-7637-3827-1.

3.9. EXTERNAL LINKS 9

[12] Mircea S. Rogalski, Stuart B. Palmer (2006). AdvancedUniversity Physics. CRC Press. p. 267. ISBN 1-58488-511-4.

[13] Stuart B. Palmer &Mircea S. Rogalski (2006). AdvancedUniversity Physics. CRC Press. p. 251. ISBN 1-58488-511-4.

[14] The magnetization current can be expressed as the curl ofthe magnetization, so its divergence is zero and it does notcontribute to the continuity equation. See magnetizationcurrent.

3.8 Further reading• Griffiths, David J. (1998). Introduction to Electro-

dynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.

• Tipler, Paul (2004). Physics for Scientists and Engi-neers: Electricity, Magnetism, Light, and ElementaryModern Physics (5th ed.). W. H. Freeman. ISBN0-7167-0810-8.

3.9 External links• Simple Nature by Benjamin Crowell Ampere’s lawfrom an online textbook

• MISN-0-138Ampere’s Law (PDF file) by KirbyMor-gan for Project PHYSNET.

• MISN-0-145 The Ampere–Maxwell Equation; Dis-placement Current (PDF file) by J.S. Kovacs forProject PHYSNET.

• The Ampère’s Law Song (PDF file) by Walter FoxSmith; Main page, with recordings of the song.

• A Dynamical Theory of the Electromagnetic FieldMaxwell’s paper of 1864

Chapter 4

Divergence theorem

“Gauss’s theorem” redirects here. For Gauss’s theoremconcerning the electric field, see Gauss’s law.“Ostrogradsky theorem” redirects here. For Ostrograd-sky’s theorem concerning the linear instability of theHamiltonian associated with a Lagrangian dependent onhigher time derivatives than the first, see Ostrogradskyinstability.

In vector calculus, the divergence theorem, also knownasGauss’s theorem orOstrogradsky’s theorem,[1][2] isa result that relates the flow (that is, flux) of a vector fieldthrough a surface to the behavior of the vector field insidethe surface.More precisely, the divergence theorem states that theoutward flux of a vector field through a closed surfaceis equal to the volume integral of the divergence over theregion inside the surface. Intuitively, it states that the sumof all sources minus the sum of all sinks gives the net flowout of a region.The divergence theorem is an important result for themathematics of engineering, in particular in electrostaticsand fluid dynamics.In physics and engineering, the divergence theorem isusually applied in three dimensions. However, it gener-alizes to any number of dimensions. In one dimension, itis equivalent to the fundamental theorem of calculus. Intwo dimensions, it is equivalent to Green’s theorem.The theorem is a special case of the more general Stokes’theorem.[3]

4.1 Intuition

If a fluid is flowing in some area, then the rate at whichfluid flows out of a certain region within that area can becalculated by adding up the sources inside the region andsubtracting the sinks. The fluid flow is represented by avector field, and the vector field’s divergence at a givenpoint describes the strength of the source or sink there.So, integrating the field’s divergence over the interior ofthe region should equal the integral of the vector field overthe region’s boundary. The divergence theorem says that

this is true.[4]

The divergence theorem is employed in any conservationlaw which states that the volume total of all sinks andsources, that is the volume integral of the divergence, isequal to the net flow across the volume’s boundary.[5]

4.2 Mathematical statement

V

Sn

nn

n

A region V bounded by the surface S = ∂V with the surface nor-mal n

Suppose V is a subset of Rn (in the case of n = 3, Vrepresents a volume in 3D space) which is compact andhas a piecewise smooth boundary S (also indicated with∂V = S ). If F is a continuously differentiable vector fielddefined on a neighborhood of V, then we have:[6]

∫∫∫V(∇ · F) dV = S (F · n) dS.

The left side is a volume integral over the volume V,the right side is the surface integral over the boundaryof the volume V. The closed manifold ∂V is quite gen-erally the boundary of V oriented by outward-pointingnormals, and n is the outward pointing unit normal fieldof the boundary ∂V. (dS may be used as a shorthand forndS.) The symbol within the two integrals stresses once

10

4.3. EXAMPLE 11

The divergence theorem can be used to calculate a flux througha closed surface that fully encloses a volume, like any of the sur-faces on the left. It can not directly be used to calculate the fluxthrough surfaces with boundaries, like those on the right. (Sur-faces are blue, boundaries are red.)

more that ∂V is a closed surface. In terms of the intuitivedescription above, the left-hand side of the equation rep-resents the total of the sources in the volume V, and theright-hand side represents the total flow across the bound-ary S.

4.2.1 Corollaries

By applying the divergence theorem in various contexts,other useful identities can be derived (cf. vector identi-ties).[6]

• Applying the divergence theorem to the product ofa scalar function g and a vector field F, the result is∫∫∫

V[F · (∇g) + g (∇ · F)] dV =

S gF · dS.

A special case of this is F = ∇ f , in which casethe theorem is the basis for Green’s identities.

• Applying the divergence theorem to the cross-product of two vector fields F × G, the result is∫∫∫

V[G · (∇× F)− F · (∇×G)] dV =

S (F×G) · dS.

• Applying the divergence theorem to the product ofa scalar function, f , and a non-zero constant vectorc, the following theorem can be proven:[7]

∫∫∫Vc · ∇f dV = S (cf) ·

dS−∫∫∫

Vf(∇ · c) dV.

• Applying the divergence theorem to the cross-product of a vector field F and a non-zero constantvector c, the following theorem can be proven:[7]

∫∫∫Vc·(∇×F) dV = S (F×

c) · dS.

4.3 Example

The vector field corresponding to the example shown. Note, vec-tors may point into or out of the sphere.

Suppose we wish to evaluate

S F · n dS,

where S is the unit sphere defined by

S ={x, y, z ∈ R3 : x2 + y2 + z2 = 1

}.

and F is the vector field

F = 2xi+ y2j+ z2k.

12 CHAPTER 4. DIVERGENCE THEOREM

The direct computation of this integral is quite difficult,but we can simplify the derivation of the result using thedivergence theorem, because the divergence theorem saysthat the integral is equal to:

∫∫∫W

(∇·F) dV = 2

∫∫∫W

(1+y+z) dV = 2

∫∫∫W

dV+2

∫∫∫W

y dV+2

∫∫∫W

z dV.

where W is the unit ball:

W ={x, y, z ∈ R3 : x2 + y2 + z2 ≤ 1

}.

Since the function y is positive in one hemisphere of Wand negative in the other, in an equal and opposite way,its total integral over W is zero. The same is true for z:

∫∫∫W

y dV =

∫∫∫W

z dV = 0.

Therefore,

S F · n dS = 2∫∫∫

WdV = 8π

3 ,

because the unit ball W has volume 4π/3.

4.4 Applications

4.4.1 Differential form and integral form ofphysical laws

As a result of the divergence theorem, a host of physi-cal laws can be written in both a differential form (whereone quantity is the divergence of another) and an inte-gral form (where the flux of one quantity through a closedsurface is equal to another quantity). Three examples areGauss’s law (in electrostatics), Gauss’s law formagnetism,and Gauss’s law for gravity.

Continuity equations

Main article: continuity equation

Continuity equations offer more examples of laws withboth differential and integral forms, related to eachother by the divergence theorem. In fluid dynamics,electromagnetism, quantum mechanics, relativity theory,and a number of other fields, there are continuity equa-tions that describe the conservation of mass, momen-tum, energy, probability, or other quantities. Generically,these equations state that the divergence of the flow of theconserved quantity is equal to the distribution of sources

or sinks of that quantity. The divergence theorem statesthat any such continuity equation can be written in a dif-ferential form (in terms of a divergence) and an integralform (in terms of a flux).[8]

4.4.2 Inverse-square laws

Any inverse-square law can instead be written in a Gauss’law-type form (with a differential and integral form, asdescribed above). Two examples are Gauss’ law (inelectrostatics), which follows from the inverse-squareCoulomb’s law, and Gauss’ law for gravity, which followsfrom the inverse-square Newton’s law of universal grav-itation. The derivation of the Gauss’ law-type equationfrom the inverse-square formulation (or vice versa) is ex-actly the same in both cases; see either of those articlesfor details.[8]

4.5 History

The theorem was first discovered by Lagrange in1762,[9] then later independently rediscovered by Gaussin 1813,[10] by Ostrogradsky, who also gave the first proofof the general theorem, in 1826,[11] by Green in 1828,[12]etc.[13] Subsequently, variations on the divergence theo-rem are correctly called Ostrogradsky’s theorem, but alsocommonly Gauss’s theorem, or Green’s theorem.

4.6 Examples

To verify the planar variant of the divergence theorem fora region R:

R ={x, y ∈ R2 : x2 + y2 ≤ 1

},

and the vector field:

F(x, y) = 2yi+ 5xj.

The boundary of R is the unit circle, C, that can be rep-resented parametrically by:

x = cos(s), y = sin(s)

such that 0 ≤ s ≤ 2π where s units is the length arc fromthe point s = 0 to the point P on C. Then a vector equationof C is

C(s) = cos(s)i+ sin(s)j.

At a point P on C:

4.8. SEE ALSO 13

P = (cos(s), sin(s)) ⇒ F = 2 sin(s)i+ 5 cos(s)j.

Therefore,

∮C

F · n ds =∫ 2π

0

(2 sin(s)i+ 5 cos(s)j) · (cos(s)i+ sin(s)j) ds

=

∫ 2π

0

(2 sin(s) cos(s) + 5 sin(s) cos(s)) ds

= 7

∫ 2π

0

sin(s) cos(s) ds

= 0.

BecauseM = 2y, ∂M/∂x = 0, and because N = 5x, ∂N/∂y= 0. Thus

∫∫R

divF dA =

∫∫R

(∂M

∂x+

∂N

∂y

)dA = 0.

4.7 Generalizations

4.7.1 Multiple dimensions

One can use the general Stokes’ Theorem to equate the n-dimensional volume integral of the divergence of a vectorfield F over a region U to the (n − 1)-dimensional surfaceintegral of F over the boundary of U:

∫U

∇ · F dVn =

∮∂U

F · n dSn−1

This equation is also known as the Divergence theorem.When n = 2, this is equivalent to Green’s theorem.When n = 1, it reduces to the Fundamental theorem ofcalculus.

4.7.2 Tensor fields

Main article: Tensor field

Writing the theorem in Einstein notation:

∫∫∫V

∂Fi

∂xidV = S Fini dS

suggestively, replacing the vector field F with a rank-ntensor field T, this can be generalized to:[14]

∫∫∫V

∂Ti1i2···iq···in∂xiq

dV = S

Ti1i2···iq···inniq dS.

where on each side, tensor contraction occurs for at leastone index. This form of the theorem is still in 3d, eachindex takes values 1, 2, and 3. It can be generalized fur-ther still to higher (or lower) dimensions (for example to4d spacetime in general relativity[15]).

4.8 See also• Stokes’ theorem

• Kelvin–Stokes theorem

4.9 Notes[1] or less correctly as Gauss' theorem (see history for rea-

son)

[2] Katz, Victor J. (1979). “The history of Stokes’s the-orem”. Mathematics Magazine (Mathematical Associa-tion of America) 52: 146–156. doi:10.2307/2690275.reprinted in Anderson, Marlow (2009). Who Gave Youthe Epsilon?: And Other Tales of Mathematical History.Mathematical Association of America. pp. 78–79. ISBN0883855690.

[3] Stewart, James (2008), “Vector Calculus”, Calculus:Early Transcendentals (6 ed.), Thomson Brooks/Cole,ISBN 978-0-495-01166-8

[4] R. G. Lerner, G. L. Trigg (1994). Encyclopaedia ofPhysics (2nd ed.). VHC. ISBN 3-527-26954-1.

[5] Byron, Frederick; Fuller, Robert (1992), Mathematics ofClassical and Quantum Physics, Dover Publications, p. 22,ISBN 978-0-486-67164-2

[6] M. R. Spiegel; S. Lipschutz; D. Spellman (2009). VectorAnalysis. Schaum’s Outlines (2nd ed.). USA: McGrawHill. ISBN 978-0-07-161545-7.

[7] MathWorld

[8] C.B. Parker (1994). McGraw Hill Encyclopaedia ofPhysics (2nd ed.). McGraw Hill. ISBN 0-07-051400-3.

[9] In his 1762 paper on sound, Lagrange treats a special caseof the divergence theorem: Lagrange (1762) “Nouvellesrecherches sur la nature et la propagation du son” (Newresearches on the nature and propagation of sound), Mis-cellanea Taurinensia (also known as: Mélanges de Turin), 2: 11 - 172. This article is reprinted as: “Nouvellesrecherches sur la nature et la propagation du son” in:J.A. Serret, ed., Oeuvres de Lagrange, (Paris, France:Gauthier-Villars, 1867), vol. 1, pages 151-316; on pages263-265, Lagrange transforms triple integrals into doubleintegrals using integration by parts.

14 CHAPTER 4. DIVERGENCE THEOREM

[10] C. F. Gauss (1813) “Theoria attractionis corporumsphaeroidicorum ellipticorum homogeneorum methodonova tractata,” Commentationes societatis regiae scientiar-ium Gottingensis recentiores, 2: 355-378; Gauss consid-ered a special case of the theorem; see the 4th, 5th, and6th pages of his article.

[11] Mikhail Ostragradsky presented his proof of the diver-gence theorem to the Paris Academy in 1826; however,his work was not published by the Academy. He returnedto St. Petersburg, Russia, where in 1828-1829 he readthe work that he'd done in France, to the St. PetersburgAcademy, which published his work in abbreviated formin 1831.

• His proof of the divergence theorem -- “Démon-stration d'un théorème du calcul intégral” (Proofof a theorem in integral calculus) -- which he hadread to the Paris Academy on February 13, 1826,was translated, in 1965, into Russian togetherwith another article by him. See: ЮшкевичА.П. (Yushkevich A.P.) and Антропова В.И.(Antropov V.I.) (1965) "Неопубликованныеработы М.В. Остроградского" (Unpublishedworks of MV Ostrogradskii), Историко-математические исследования (Istoriko-Matematicheskie Issledovaniya / Historical-Mathematical Studies), 16: 49-96; see the sectiontitled: "Остроградский М.В. Доказательствоодной теоремы интегрального исчисления"(Ostrogradskii M. V. Dokazatelstvo odnoy teoremyintegralnogo ischislenia / Ostragradsky M.V. Proofof a theorem in integral calculus).

• M. Ostrogradsky (presented: November 5, 1828 ;published: 1831) “Première note sur la théorie dela chaleur” (First note on the theory of heat) Mé-moires de l'Académie impériale des sciences de St.Pétersbourg, series 6, 1: 129-133; for an abbrevi-ated version of his proof of the divergence theorem,see pages 130-131.

• Victor J. Katz (May1979) “The history of Stokes’theorem,” Mathematics Magazine, 52(3): 146-156;for Ostragradsky’s proof of the divergence theorem,see pages 147-148.

[12] George Green, An Essay on the Application of Mathemat-ical Analysis to the Theories of Electricity and Magnetism(Nottingham, England: T. Wheelhouse, 1838). A formof the “divergence theorem” appears on pages 10-12.

[13] Other early investigators who used some form of the di-vergence theorem include:

• Poisson (presented: February 2, 1824 ; published:1826) “Mémoire sur la théorie du magnétisme”(Memoir on the theory of magnetism), Mémoiresde l'Académie des sciences de l'Institut de France, 5:247-338; on pages 294-296, Poisson transforms avolume integral (which is used to evaluate a quan-tity Q) into a surface integral. To make this trans-formation, Poisson follows the same procedure thatis used to prove the divergence theorem.

• Frédéric Sarrus (1828) “Mémoire sur les oscilla-tions des corps flottans” (Memoir on the oscillations

of floating bodies), Annales de mathématiques pureset appliquées (Nismes), 19: 185-211.

[14] K.F. Riley, M.P. Hobson, S.J. Bence (2010). Mathemati-cal methods for physics and engineering. Cambridge Uni-versity Press. ISBN 978-0-521-86153-3.

[15] see for example:J.A. Wheeler, C. Misner, K.S. Thorne (1973).Gravitation. W.H. Freeman & Co. pp. 85–86,§3.5. ISBN 0-7167-0344-0., andR. Penrose (2007). The Road to Reality. Vintage books.ISBN 0-679-77631-1.

4.10 External links• Hazewinkel, Michiel, ed. (2001), “Ostrogradskiformula”, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• Differential Operators and the Divergence Theoremat MathPages

• The Divergence (Gauss) Theorem by Nick Bykov,Wolfram Demonstrations Project.

• Weisstein, Eric W., “Divergence Theorem”,MathWorld. — This article was originallybased on the GFDL article from PlanetMath athttp://planetmath.org/encyclopedia/Divergence.html

Chapter 5

Electric displacement field

In physics, the electric displacement field, denoted byD, is a vector field that appears in Maxwell’s equations. Itaccounts for the effects of free and bound charge withinmaterials. "D" stands for “displacement”, as in the re-lated concept of displacement current in dielectrics. Infree space, the electric displacement field is equivalent toflux density, a concept that lends understanding to Gauss’slaw. It has the SI units of coulomb per squared metre (Cm−2).

5.1 Definition

In a dielectric material the presence of an electric field Ecauses the bound charges in the material (atomic nucleiand their electrons) to slightly separate, inducing a localelectric dipole moment. The electric displacement fieldD is defined as

D ≡ ε0E+ P,

where ε0 is the vacuum permittivity (also called permit-tivity of free space), and P is the (macroscopic) densityof the permanent and induced electric dipole momentsin the material, called the polarization density. Separat-ing the total volume charge density into free and boundcharges:

ρ = ρf + ρb

the density can be rewritten as a function of the polariza-tion P:

ρ = ρf −∇ · P.

The polarization P is defined to be a vector field whosedivergence yields the density of bound charges ρ in thematerial. The electric field satisfies the equation:

∇ · E =1

ε0ρ =

1

ε0(ρf −∇ · P)

and hence

∇ · (ε0E+ P) = ρf

The displacement field therefore satisfies Gauss’s law in adielectric:

∇ · D = ρ− ρb = ρf

However, electrostatic forces on ions or electrons in thematerial are still governed by the electric field E in thematerial via the Lorentz Force. It should also be remem-bered that D is not determined exclusively by the freecharge. Consider the relationship:

∇× D = ε0∇× E+∇× P,

which, by the fact that E has a curl of zero in electrostaticsituations, evaluates to:

∇× D = ∇× P

The effect of this equation can be seen in the case of anobject with a “frozen in” polarization like a bar electret,the electric analogue to a bar magnet. There is no freecharge in such a material, but the inherent polarizationgives rise to an electric field. If the wayward student wereto assume that the D field were entirely determined bythe free charge, he or she would conclude that the electricfield were zero outside such a material, but this is patentlynot true. The electric field can be properly determinedby using the above relation along with other boundaryconditions on the polarization density to yield the boundcharges, which will, in turn, yield the electric field.In a linear, homogeneous, isotropic dielectric with instan-taneous response to changes in the electric field, P de-pends linearly on the electric field,

P = ε0χE,

15

16 CHAPTER 5. ELECTRIC DISPLACEMENT FIELD

where the constant of proportionality χ is called theelectric susceptibility of the material. Thus

D = ε0(1 + χ)E = εE

where ε = ε0 εᵣ is the permittivity, and εᵣ = 1 + χ therelative permittivity of the material.In linear, homogeneous, isotropic media, ε is a constant.However, in linear anisotropic media it is a tensor, andin nonhomogeneous media it is a function of position in-side the medium. It may also depend upon the electricfield (nonlinear materials) and have a time dependent re-sponse. Explicit time dependence can arise if the ma-terials are physically moving or changing in time (e.g.reflections off a moving interface give rise to Dopplershifts). A different form of time dependence can arisein a time-invariant medium, in that there can be a timedelay between the imposition of the electric field and theresulting polarization of the material. In this case, P is aconvolution of the impulse response susceptibility χ andthe electric field E. Such a convolution takes on a simplerform in the frequency domain—by Fourier transform-ing the relationship and applying the convolution theo-rem, one obtains the following relation for a linear time-invariant medium:

D(ω) = ε(ω)E(ω),

where ω is the frequency of the applied field. The con-straint of causality leads to the Kramers–Kronig relations,which place limitations upon the form of the frequencydependence. The phenomenon of a frequency-dependentpermittivity is an example of material dispersion. Infact, all physical materials have some material dispersionbecause they cannot respond instantaneously to appliedfields, but for many problems (those concerned with anarrow enough bandwidth) the frequency-dependence ofε can be neglected.At a boundary, (D1 − D2) · n̂ = D1,⊥ − D2,⊥ = σf ,where σ is the free charge density and the unit normal n̂points in the direction from medium 2 to medium 1.[1]

5.2 History

Recall that Gauss’s law was formulated by Carl FriedrichGauss in 1835, but was not published until 1867. Thisplaces a lower limit on the year of the formulation anduse of D to not earlier than 1835. Again, considering thelaw in its usual form using E vector rather than D vector,it can be safely said thatD formalism was not used earlierthan 1860s.The Electric Displacement field was first found to be usedin the year 1864 by James Clerk Maxwell, in his pa-per A Dynamical Theory of the Electromagnetic Field.

Maxwell used calculus to exhibit Michael Faraday’s the-ory, that light is an electromagnetic phenomenon. InPART V. — THEORY OF CONDENSERS, Maxwellintroduced the term D in page 494, titled, Specific Ca-pacity of Electric Induction (D), explained in a form dif-ferent from the modern and familiar notations.Confusion over the term “Maxwell’s equations” arises be-cause it has been used for a set of eight equations that ap-peared in Part III of Maxwell’s 1864 paper A DynamicalTheory of the Electromagnetic Field, entitled “GeneralEquations of the Electromagnetic Field”, and this confu-sion is compounded by the writing of six of those eightequations as three separate equations (one for each of theCartesian axes), resulting in twenty equations and twentyunknowns. (As noted above, this terminology is not com-mon: Modern references to the term “Maxwell’s equa-tions” refer to the Heaviside restatements.)It was Oliver Heaviside who reformulated the compli-cated Maxwell’s equations to the modern, elegant formthat we know today. But it wasn't until 1884 that Heav-iside, concurrently with similar work by Willard Gibbsand Heinrich Hertz, grouped them together into a distinctset. This group of four equations was known variouslyas the Hertz–Heaviside equations and theMaxwell–Hertzequations, and are sometimes still known as theMaxwell–Heaviside equations.Hence, it was probably Heaviside who lent D the presentsignificance it now has.[2]

5.3 Example: Displacement field ina capacitor

A parallel plate capacitor. Using an imaginary pillbox, it is possi-ble to use Gauss’s law to explain the relationship between electricdisplacement and free charge.

Consider an infinite parallel plate capacitor placed inspace (or in a medium) with no free charges present ex-cept on the capacitor. In SI units, the charge density onthe plates is equal to the value of the D field betweenthe plates. This follows directly from Gauss’s law, by in-tegrating over a small rectangular pillbox straddling one

5.5. REFERENCES 17

plate of the capacitor:

∮A

D · dA = Qfree

On the sides of the pillbox, dA is perpendicular to thefield, so that part of the integral is zero, leaving, for thespace inside the capacitor where the fields of the twoplates add,

|D| = QfreeA

where A is surface area of the top face of the small rect-angular pillbox and Q ᵣₑₑ / A is just the free surface chargedensity on the positive plate. Outside the capacitor, thefields of the two plates cancel each other and |E| = |D| =0. If the space between the capacitor plates is filled witha linear homogeneous isotropic dielectric with permittiv-ity ε, the total electric field E between the plates will besmaller than D by a factor of ε: |E| = Q ᵣₑₑ / (εA).If the distance d between the plates of a finite parallelplate capacitor is much smaller than its lateral dimensionswe can approximate it using the infinite case and obtainits capacitance as

C =QfreeV

≈ Qfree|E|d =

A

dε,

where V is the potential difference sustained between thetwo plates. The partial cancellation of fields in the dielec-tric allows a larger amount of free charge to dwell on thetwo plates of the capacitor per unit potential drop thanwould be possible if the plates were separated by vacuum.

5.4 See also• History ofMaxwell’s equations#The termMaxwell’sequations

• Polarization density

• Electric susceptibility

• Magnetizing field

• Electric dipole moment

5.5 References[1] David Griffiths. Introduction to Electrodynamics (3rd

1999 ed.).

[2] History of Maxwell’s equations#The term Maxwell’sequations

• “electric displacement field” at PhysicsForums

18 CHAPTER 5. ELECTRIC DISPLACEMENT FIELD

5.6 Text and image sources, contributors, and licenses

5.6.1 Text

• Gauss’s law Source: https://en.wikipedia.org/wiki/Gauss’{}s_law?oldid=680155183 Contributors: The Anome, Nd12345, Hephaestos,Patrick, Michael Hardy, Ixfd64, TakuyaMurata, Delirium, Looxix~enwiki, Ahoerstemeier, LittleDan, AugPi, Charles Matthews, Xaven,Lambda, Covracer, Fuelbottle, Diberri, Giftlite, Andries, Lethe, Koyn~enwiki, Tagishsimon, MFNickster, Discospinster, Goplat, El-wikipedista~enwiki, JustinWick, Laurascudder, Spoon!, Rbj, Emhoo~enwiki, Deryck Chan, LutzL, Wricardoh~enwiki, Keenan Pep-per, ABCD, Versageek, Linas, StradivariusTV, Insurrectionist, Pfalstad, Rnt20, Rjwilmsi, JHMM13, The wub, Drrngrvy, Margos-bot~enwiki, Nickpowerz, Chobot, DVdm, WriterHound, Roboto de Ajvol, YurikBot, Cookie4869~enwiki, Cpl.Luke, Salsb, IlyaV,Irishguy, Blu Aardvark, Mikeblas, E2mb0t~enwiki, Gadget850, DeadEyeArrow, Bota47, NBS525, Rdrosson, Sbyrnes321, SmackBot, RD-Bury, Bigbluefish, MonteChristof, Thumperward, DHN-bot~enwiki, TheKMan, Amazins490, Daniel.Cardenas, Yevgeny Kats, DJIndica,Nmnogueira, Pinkbasu, Cronholm144, Slicerace, Scetoaux, Achoo5000, Chetvorno, Merryjman, Mosaffa, MikeMalak, Physic sox, CBM,Ali Obeid, Frozen fish, Cydebot, Reywas92, Christian75, Dchristle, Thijs!bot, Irigi, Headbomb, Pjvpjv, Eleuther, AntiVandalBot, JAnDbot,Retinoblastoma, Random89, PhilKnight, JamesBWatson, HebrewHammerTime, Patstuart, Dr. Morbius, Mcba, Lcabanel, Social tamarisk,Enok.cc, Yonidebot, Bcartolo, LordAnubisBOT, SmilesALot, 2help, Treisijs, Adanadhel, Lseixas, Yecril, Izno, Sheliak, Quiet Silent Bob,VolkovBot, Benjamin Barenblat, Lingwitt, TXiKiBoT, Oshwah, One zero one, IAPIAP, Aihadley, SieBot, BotMultichill, Gerakibot, Cal-tas, Kojiroh, Dahnamics, ClueBot, Delver23, DragonBot, Kocher2006, PixelBot, Rhododendrites, Brews ohare, Eebster the Great, Lifeof Riley, Badgernet, Alexius08, MystBot, Addbot, Kuri ahf, Mphilip1, SamatBot, Jasper Deng, Ikara, Firoja, Tide rolls, Lightbot, Zor-robot, Legobot, Luckas-bot, Yobot, Ptbotgourou, KamikazeBot, Rama1981, Aa77zz, GrouchoBot, Destatiforze, Imveracious, BoomerAB,Rupeld, Edisonorellana12, Meier99, Vrenator, Miracle Pen, Onel5969, Jmannc3, TjBot, Set theorist, Solomonfromfinland, Anir1uph,Quondum, SporkBot, Donner60, Puffin, ClamDip, ClueBot NG, MelbourneStar, Masssly, ساجد امجد ,ساجد Helpful Pixie Bot, Vagobot,JacDT, Ibelieveilove, F=q(E+v^B), United States Man, YFdyh-bot, IngenieroLoco, Lugia2453, Andyhowlett, K7hoven, Ismayil.rahmaniya,Mirza Sakić, Shabab Rahman, SantiLak, TerryAlex, Hasanb07, Shreyas gonjari, SilverSylvester and Anonymous: 186

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